Mechanical Systemsand

Signal ProcessingMechanical Systems and Signal Processing 21 (2007) 28142832

Although this model is useful for frequency domain analysis, it can be difcult to implement in the time

ARTICLE IN PRESS

www.elsevier.com/locate/jnlabr/ymssp0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2007.02.007

Corresponding author. Tel.: +1 502 852 6336; fax: +1 502 852 6053.E-mail address: chris.richards@louisville.edu (C.M. Richards).1. Introduction

The elastomeric isolator is widely used in noise and vibration control as means of isolating vibrationsources. For a successful prediction of dynamic behavior of an elastomeric isolation system subject to a givenexcitation, it is important to have an accurate mathematical model of the isolator. The Voigt model (Fig. 1(a)),which is a spring and damper in parallel, is often used for modeling elastomeric isolators owing to itssimplicity in analysis and parameter identication [1]. However, dynamic stiffness experiments in which therubber isolator is subjected to sinusoidal deformations often reveal the frequency-dependent features ofelastomeric isolators, making the Voigt model inaccurate [2,3]. The frequency-dependent complex stiffnessmodel is an approach which allows the complex stiffness to vary as a function of excitation frequency [35].Abstract

A parameter identication method based on constraint optimization is developed for a single mass elastomeric isolation

system where the isolator is represented by a Maxwell model with two or more Maxwell elements. The method utilizes

measured static stiffness and frequency response of the isolator in a single mass conguration with constraints on the

natural frequency and damping ratio. It is revealed through analytical examples that Maxwell models consisting of only

one or two Maxwell elements can accurately replicate the dynamic behavior of Maxwell systems having two or more

Maxwell elements. To experimentally evaluate the method, three different rubber isolators are considered. For all three

rubber isolators, it is shown that Voigt models are incapable of accurately representing the measured static stiffness and

frequency response. Although identied Maxwell models having only one Maxwell element can match the measured

natural frequency, damping ratio and static stiffness, they cannot match the measured frequency response curves well.

However, identied Maxwell models with two Maxwell elements can accurately represent the measured static and dynamic

characteristics of the real elastomeric isolation systems.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Rubber isolator; Maxwell model; Parameter identicationParameter identication of analytical and experimental rubberisolators represented by Maxwell models

Jie Zhang, Christopher M. Richards

Department of Mechanical Engineering, The University of Louisville, Louisville, KY 40292, USA

Received 19 June 2006; received in revised form 31 January 2007; accepted 10 February 2007

Available online 2 March 2007

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2815domain [6]. The Maxwell model is a linear time domain model for elastomeric material [7]. With a series ofVoigt models and Maxwell models in series or parallel, the combination can simulate viscoelastic properties ofmany different materials.To address the nonlinear viscoelastic behavior of rubber isolators, nonlinear models are often identied and

utilized [810]. Systematic experiments have been conducted in both single and multi-degree-of-freedomcongurations while static, random, frequency-sweep, and xed frequency excitations were applied where thenonlinear dynamic behavior of elastomeric isolators was addressed [11]. Due to its versatility, Bouc-Wendifferential models are often used for representing friction-type hysteretic isolators [12]. A parameteridentication method based on least squares estimation in the frequency domain is developed for theBouc-Wen model using experimental data from periodic vibration experiments. A bilinear hysteretic model isan alternative method for hysteretic isolators [13]. The parameters are identied by minimizing the discrepancybetween the measured responses and the theoretical responses of the system in the time domain. Theidentication of nonlinear isolators using temporal and spectral methods have been investigated [14]. It wasfound that non-integer exponent-type terms were best for describing the nonlinear elastic force of the rubber.The equivalent linearization technique has also been developed to linearize the governing nonlinear multi-degree-of-freedom equations of motion where responses of interest are calculated from the linearizedgoverning equations of motion [15]. Nonetheless, it is often a challenge to determine a unique model andparameter identication process that yields an accurate model for dynamic analysis of elastomeric isolators formany different loading conditions. The model which uses fractional derivatives has been investigated bymeans of experiments and modeling [16]. The model was found superior to the Voigt model when modelingthe dynamic properties of rubber isolators.In this article a parameter identication method is developed for Maxwell models having two or more

Maxwell elements in a single mass isolation system (Fig. 1(b)) by tting the model to measured frequency

m

k1

c1

k c

x

x1

f

kn

cn

xn

k c

x fm

Fig. 1. Mathematic model of isolator represented by (a) Voigt model; (b) Maxwell model.response spectrum by means of constraint optimization. Studies conducted with analytical systems reveal thata Maxwell model having only one Maxwell element can simulate the dynamic characteristics of a Maxwellsystem having two Maxwell elements if they do not belong to a specic combination of Maxwell elementTypes. However, further results show that a Maxwell model having two Maxwell elements can simulate thedynamic characteristics of analytical Maxwell systems having three or more Maxwell elements of differentTypes.To examine the Maxwell model identication approach experimentally, three different rubber isolators are

subjected to both static and dynamic excitations. It is shown that a Voigt model is incapable of accuratelymodeling the static and dynamic characteristics of these isolators. Likewise, Maxwell models having only oneMaxwell element can be identied to have the same natural frequency, damping ratio, static stiffness, of theisolators in a single mass conguration; however, the frequency response of these models do not match themeasured frequency response spectra well. By using the approach developed in this article, Maxwellmodels having two Maxwell elements are identied that have the same natural frequency, damping ratio andstatic stiffness of the isolators in a single mass conguration, while accurately replicating their measuredfrequency response.

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 2814283228162. Problem formulation

The governing equations of motion for the Maxwell system represented in Fig. 1(b) are

m 0 0 00 0 0 00 0 0 0... ..

. ... . .

. ...

0 0 0 0

2666666664

3777777775

x

x1

x2

..

.

xn

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

c 0 0 00 c1 0 00 0 c2 0... ..

. ... . .

. ...

0 0 0 cn

26666666664

37777777775

_x

_x1

_x2

..

.

_xn

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

k k1 kn k1 k2 knk1 k1 0 0k2 0 k2 0

..

. ... ..

. . .. ..

.

kn 0 0 kn

26666666664

37777777775

x

x1

x2

..

.

xn

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

f

0

0

..

.

0

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

. 1

Writing Eq. (1) in statespace form

x

_x

_x1

_x2

..

.

_xn

8>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

cm

k k1 knm

k1

m

k2

m kn

m

1 0 0 0 0

01

t1 1t1

0 0

01

t20 1

t2 0

..

. ... ..

. ... . .

. ...

01

tn0 0 1

tn

266666666666666666664

377777777777777777775

_x

x

x1

x2

..

.

xn

8>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

f 0

0

0

0

..

.

0

8>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

, 2

where f 0 f =m, ti ci=ki is the time constant of each Maxwell element and n is the number of Maxwellelements in the model. The modal analysis is based on the eigenvalue problem of the n 2 by n 2 systemmatrix in Eq. (2), which yields n real and one complex conjugate pair of eigenvalues when the system is under-damped. For the purpose of single mass system vibration analysis, only the conjugated eigenvalues are ofconcern from which the natural frequency on and damping ratio z of the system are determined.In order for a Maxwell model, as given in Eq. (1) or (2), to accurately represent a real physical isolation

mount, the parameters of the model must be identied using test data collected from experiments conductedon the mount. For example, the isolated mass m can be measured on a weight scale and the primary linear

spring stiffness k can be estimated from a static stiffness experiment. Therefore, c, ci and ki; i 2 1; n are

parameters left to be determined. Zhang and Richards [17] propose a method appropriate when only oneMaxwell element exists. However, as illustrated in the following section, although a Maxwell model with oneMaxwell element is accurate for certain types of systems, the model is inaccurate for others. Therefore, a newmethod based on constraint optimization is developed in this article for identifying Maxwell models consistingof more than one Maxwell element. The method is validated using theoretical models (Section 4) and thenused to identify three physical isolators (Section 6).

3. Identication of Maxwell systems using MV models

The parameter identication method developed by Zhang and Richards [17] is appropriate when onlyone Maxwell element exists in the model (i.e., MV models). The method is described in detail in Ref. [17]but will be given in brevity here for completeness. Since the isolated mass m can be measured on a weightscale and the primary linear spring stiffness k can be estimated from a static stiffness experiment, theparameters c, c1 and k1 are left to be determined. By estimating omv and zmv from the measured frequency

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2817response by experimental modal analysis and assuming a value for c, then constant omv and zmv curves of themodel can be drawn on a plane with coordinates of c1 and k1. The specic values for c1 and k1 that yield omvand zmv for the MV model are then identied by the intersection point of the constant omv and zmv curves(Fig. 2). However, the constant omv and zmv curves are based on an assumed value for c. Therefore, differentmodels are generated by assuming different values of c accompanied with unique values for c1 and k1. Theidentied model is then chosen by comparing each models frequency response with the measured frequencyresponse of the system.Unfortunately, an MV model does not always satisfactorily provide an adequate t to the data when the

system being identied consists of more than one Maxwell element. To illustrate this, six different Maxwellsystems with two Maxwell elements (MMV systems) are considered. Note, the term system as used in thisarticle refers to the experimental system to be modeled. It is assumed that the systems isolated mass m canbe measured on a weight scale, the primary linear spring stiffness k can be estimated from a static stiffnessexperiment, and the frequency response is available by means of vibration experiment. The term modelrefers to the analytical Maxwell model whose parameters are identied from the systems frequencyresponse. All parameters of the six systems considered are listed in Table 1. For different ranges of values forti, three unique types of Maxwell elements are dened [17]. In general, for small values of ti, the springstiffness is so great that motion primarily occurs across the damper. This type of the Maxwell element isdamping dominant and is dened as a Type A element. For large values of ti, the damping is so great thatmotion primarily occurs across the spring. This type of the Maxwell element is stiffness dominant and isdened as a Type C element. For intermediate values of ti, motion across one element does not dominate overmotion across the other element. Therefore, this type of the Maxwell element is neither damping nor stiffnessdominant and is dened as a Type B element.Fig. 2. Identifying k1 and c1 from constant omv and zmv curves. constant omv curves; constant zmv curves.

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322818Table 2

Parameters of identied MV models

Parameters M V Model A M V Model B M V Model C M V Model D M V Model E M V Model F

c (N s/m) 49.702 40.382 39.724 58.030 60.637 61.395

k1 (kN/m) 18.788 35.492 35.987 12.695 12.420 33.221

c1 (N s/m) 30.447 196.437 3476.033 83.162 2072.516 1482.569

t1 (ms) [Type] 1.62 [A] 5.53 [B] 96.59 [C] 6.55 [B] 166.87 [C] 44.63 [C]

Table 1

Parameters of actual MMV Systems

Parameters M M V

System A

M M V

System B

M M V

System C

M M V

System D

M M V

System E

M M V

System F

m (kg) 1 1 1 1 1 1

k (kN/m) 12 12 12 12 12 12

c (N s/m) 40 40 40 40 40 40

k1 (kN/m) 12 12 12 24 24 12

c1 (N s/m) 20 80 2000 20 20 80

t1 (ms) [Type] 1.67 [A] 6.67 [B] 166.67 [C] 1.67 [A] 1.67 [A] 6.67 [B]k2 (kN/m) 24 24 24 12 12 24

c2 (N s/m) 20 120 2000 80 2000 2000

t2 (ms) [Type] 0.83 [A] 5.00 [B] 83.33 [C] 6.67 [B] 166.67 [C] 83.33 [C]on (rad/s) 112.684 216.050 219.718 152.126 156.833 214.806z 0.371 0.404 0.108 0.432 0.203 0.182For the examples in Table 1, there are two Type A Maxwell elements in System A, two Type B Maxwellelements in System B, two Type C Maxwell elements in System C, one Type A and one Type B Maxwellelement in System D, one Type A and one Type C Maxwell element in System E, and one Type B and one TypeC Maxwell element in System F. From the frequency response spectra of the actual MMV systems, theparameter identication method in Ref. [17] identies these systems as MV models (i.e., having only a singleMaxwell element). The resulting parameters of identied MV models are listed in Table 2, and the frequencyresponse spectra of the actual MMV systems and identied MV models are shown in Fig. 3.As illustrated in Fig. 3(a)(c), the frequency response curves of the identied MV models closely match

those of the corresponding MMV systems. The Maxwell element is Type A in identied MV Model A, TypeB in identied M V Model B and Type C in identied M V Model C. This illustrates that MV models canbe accurately identied for MMV systems if the two Maxwell elements making up the systems are of thesame Type.Since the Type A elements in M M V System D and E are damping dominant, the effective damping of

these systems is approximately the sum c c1. Consequently, the identication process identies the primarydampers of the corresponding MV models as approximately this sum. Also, the Maxwell element of the M VModel D is identied as Type B and the Maxwell element of the M V Model E is identied as Type C, whichare consistent types compared to the non-Type A Maxwell elements making up M M V System D and E.Figs. 3(d)(e) comparing the frequency response curves of the identied M V Model D and E to those of theM M V System D and E, respectively, illustrate that since damping dominant elements exist in M M VSystem D and E, the identied MV models can accurately represent these systems.Unfortunately, the frequency response spectrum of identied M V Model F does not accurately match the

actual M M V System F (Fig. 3(f)). Although the Type C element of the M M V System F is stiffnessdominant, the primary spring stiffness k of the model is xed since its value is determined from static stiffnessexperiment. Also, no single Type B or Type C Maxwell element can simulate the combined behavior of a TypeB and Type C element. Consequently, a parameter identication method for Maxwell models having two ormore Maxwell elements is necessary.

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 28190 50 100 200 250 300

0

0 50 100 200 250 300

0

0.08

magnitude (

mm

/N)

magnitude (

mm

/N)

0.2

0.15

0.1

0.05

frequency (rad/s)

150

frequency (rad/s)

150

0.1

0.06

0.04

0.02

a b4. Parameter identication using constraint optimization

The parameter identication process for determining the 2n 1 unknowns c; c1; . . . ; cn and k1; . . . ; kn ofMaxwell models with n Maxwell elements is based on the nonlinear constraint optimization problem.A sequential quadratic programming (SQP) method is used as the solution algorithm for the identication[18]. The objective function is

f k XNj1

absHjk Hj ! min for k 2 O2n1, (3)

where N is the number of discrete frequencies of the measured complex frequency response Hj. The vector kcontains the 2n 1 unknown parameters of the Maxwell model governed by Eq. (1) and Hjk are calculated

magnitude (

mm

/N)

0

magnitude (

mm

/N)

magnitude (

mm

/N)

magnitude (

mm

/N)

0.1

0.08

0.06

0.04

0.02

0.12

0.1

0.08

0.06

0.04

0.02

0.06

0.04

0.02

0.08

0.1

0.1

0.08

0.06

0.04

0.02

0

0 50 100 200 250 300 0 50 100 200 250 300

frequency (rad/s)

150

frequency (rad/s)

150

0 50 100 200 250 300 0 50 100 200 250 300

frequency (rad/s)

150

frequency (rad/s)

150

c d

e f

Fig. 3. Frequency response of MMV systems and corresponding identied MV models (a) System and Model A; (b) System and Model

B; (c) System and Model C; (d) System and Model D; (e) System and Model E; (f) System and Model F.

complex frequency response data of the model at the discrete frequencies. The optimization problem Eq. (3)minimizes the objective function f(k), where the parameters k are subjected to the following constraints:

onk on2 0 (4a)and

zk z2 0, (4b)where onk and zk are the calculated natural frequency and damping ratio governed by Eq. (2), and on andz are estimated from the measured frequency response.To illustrate the effectiveness of the parameter identication using the optimization method, the parameter

identication of the previous M M V System F is considered. To simulate measured frequency responsedata, random noise is added:

Hj Hjk0 ajHjk0, (5)where aj is a normally distributed variable with zero mean and standard deviation s 0, 0.01 and 0.05. Thevector k0 consists of the parameters of the actual M M V System F. During the identication, a large rangebetween the lower and upper bounds on the parameters is applied and the initial estimates are chosen very faraway from the real parameters. But, the identifying process is quick and the results are satised.The parameters of the identied M M V Model F are listed in Table 3, which match closely to the

parameters of the actual M M V System F in the case of s 0 and 0.01. Although the results are different incase of s 0.05, the frequency response spectrum of identied M M V Model F is very close to that ofactual MMV System F (Fig. 4).Now consider two Maxwell systems in Table 4. There are three Maxwell elements in the Maxwell System A

and six Maxwell elements in Maxwell System B. The optimization method is used to identify these Maxwell

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322820Table 3

Parameters of identied MMV Model F

s 0 0.01 0.05

c (N s/m) 39.9903 39.9552 43.5938

k1 (kN/m) 12.0025 11.9991 11.3901

c1 (N s/m) 79.9205 79.7038 86.9666

k2 (kN/m) 24.0041 24.0201 23.8265

c2 (N s/m) 1999.97 1998.76 1993.82

0 10 20 30 40 50

0

magnitude (

mm

/N)

0.1

0.08

0.06

0.04

0.02

frequency (Hz)Fig. 4. Frequency response spectra of actual MMV System F and identied MMV Model F s 0:05.

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2821Table 4

Parameters of Maxwell systems

General Maxwell System A General Maxwell System B

m (kg) 1 m (kg) 1 k4 (kN/m) 24

k (kN/m) 12 k (kN/m) 12 c4 (N s/m) 120

c (N s/m) 40 c (N s/m) 40 t4 (ms) [Type] 5.00 [B]k1 (kN/m) 12 k1 (kN/m) 12 k5 (kN/m) 12

c1 (N s/m) 20 c1 (N s/m) 20 c5 (N s/m) 2000

t1 (ms) [Type] 1.67 [A] t1 (ms) [Type] 1.67 [A] t5 (ms) [Type] 166.67 [C]k2 (kN/m) 12 k2 (kN/m) 24 k6 (kN/m) 24

c2 (N s/m) 80 c2 (N s/m) 20 c6 (N s/m) 2000

t2 (ms) [Type] 6.67 [B] t2 (ms) [Type] 0.83 [A] t6 (ms) [Type] 83.33 [C]k3 (kN/m) 24 k3 (kN/m) 12 on (rad/s) 293.8688c3 (N s/m) 2000 c3 (N s/m) 80 Z 0.2617

t3 (ms) [Type] 83.33 [C] t3 (ms) [Type] 6.67 [B] on (rad/s) 112.6844 z 0.3710

Table 5

Parameters of identied Maxwell models

General Maxwell Model A General Maxwell Model Bsystems as MMV models (i.e., having only two Maxwell elements). The resulting identied parameters ofthe models are listed in Table 5. In both identied MMV models, the resulting Maxwell elements are Type Band Type C. This is consistent with the results of Section 3, i.e., since Type A elements are damping dominant,the combination of Type A elements and the primary damper can be replaced by a new single primary damperwithout signicant modeling error. Therefore, it is not necessary to include Type A elements in the models.However, since the primary spring is determined by the static stiffness, and this stiffness along with the knownmass may not result in the correct natural frequency, stiffness dominant Type C elements are necessaryto correct for this discrepancy. Also, since Type B elements are neither damping nor stiffness dominant,these elements have a special effect that can only be replicated by Type B elements. The frequency responsespectra of the identied MMV models match the frequency response spectra of actual Maxwell systemsvery well (Fig. 5).The identied models are not unique since the identied parameters vary depending on the initial guesses

chosen. However, accurate estimates of the frequency response can be achieved if the following two rules areutilized when selecting the initial guesses. First, the initial guesses for the elements of the Maxwell modelshould be chosen so that the resulting element Types are not the same, e.g., choose one element to be Type Band one Type C. Second, the initial guesses of the elements should be chosen so that the natural frequency anddamping ratio of the Maxwell model are close to that of the actual system. Curves showing the inuences of ciand ki on natural frequency and damping ratio found in Ref. [17] are helpful in choosing initial guesses.Although it is difcult to know exactly how many Maxwell elements are necessary to identify an actualMaxwell system, these examples illustrate that it is possible to identify a Maxwell model with only twoMaxwell elements that can accurately simulate the static and dynamic behavior of a Maxwell system havingmore than two Maxwell elements.

c (N s/m) 49.5372 c (N s/m) 65.7334

k1 (kN/m) 14.9486 k1 (kN/m) 40.6862

c1 (N s/m) 81.7606 c1 (N s/m) 209.1461

t1 (ms) [Type] 5.47 [B] t1 (ms) [Type] 5.14 [B]k2 (kN/m) 24.6481 k2 (kN/m) 36.4962

c2 (N s/m) 2003.24 c2 (N-s/m) 3797.60

t2 (ms) [Type] 81.27 [C] t2 (ms) [Type] 104.05 [C]

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322822magnitude (

mm

)

0.1

0.08

0.06

0.04

0.02

a5. Experimental setup and results

A bubble mount (Fig. 6(a)), plate mount (Fig. 6(b)) and rubber stud (Fig. 6(c)) are considered for theexperimental portion of this study. In practice, these mounts are used to isolate vibration and shock inelectronic or medical equipment, avionics, computers, small pumps, compressors, appliances, ofce machinesand transportation equipment [19].

5.1. Static stiffness experiment

The static stiffness experiment employed to determine the primary stiffness k of the isolators is shown inFig. 7. The deection is manually adjusted in 0.25mm increments for the bubble mount, 0.1mm increments

0 10 20 30 40 50

0

0 10 20 30 40 50

0

magnitude (

mm

)

frequency (Hz)

0.1

0.08

0.06

0.04

0.02

frequency (Hz)

b

Fig. 5. The frequency response spectra of (a) Maxwell System A and identied M M V Model A; (b) Maxwell System B and identied

M M V Model B.

Fig. 6. (a) Bubble mount; (b) plate mount; (c) rubber stud.

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2823for the plate mount and 0.05mm increments for the rubber stud. The forces acting on load cell (connected inseries with the mounts) are read after the deections are xed for 1min. Both the loading and unloadingprocesses are measured and the resulting load versus deection curves along with the directions of bothloading and unloading are shown in Fig. 8. The bubble mount exhibits a softening spring behavior for smalldeections, but exhibits a hardening spring behavior for large deections. The plate mount behavesapproximately linear and the rubber stud exhibits a hardening spring behavior over the deection rangesconsidered. The mass used in the frequency response experiment (Section 5.2) compresses each mount to astatic equilibrium position xoE0.75, 0.5 and 0.2mm for the bubble mount, plate mount and rubber stud,respectively. An average static stiffness for each mount about xo under the loading and unloading processes istaken within the dynamic displacement limits of the frequency response experiment by the least means squaremethod. The static stiffness results are listed in Table 6.

5.2. Frequency response experiment

Fig. 9 contains a photograph and schematic of the experimental setup for measuring frequency response ofeach isolator mounted to a cylindrical block of mass m 1:041 kg. A drill press is used to provide a rigidfoundation for the experiment, as shown in Fig. 9(a). To maintain a balanced mass and minimize rockingmotion, three accelerometers are used. Fixed frequency sine excitations are applied in 0.5Hz increments from3 to 80Hz for the bubble and plate mount and from 3 to 160Hz for the rubber stud. The measured frequencyresponse spectra shown in Fig. 10 result from frequency domain average response spectra of the threeaccelerometers. The natural frequencies are determined from zero crossings of the real part of the measuredfrequency response spectra and the damping ratios are estimated by the half-power point method [20]. Resultsare listed in Table 7.

6. Parameter identication

Fig. 7. Static stiffness experiment system.The parameter identication methods developed in Ref. [17] and in this article are used to identify therubber isolators as Voigt, MV and MMV models. The accuracy of each model is illustrated by comparingthe frequency responses of each identied model with the corresponding measured frequency responses fromthe experiment.

6.1. Voigt model parameter identification

Since on and z are estimated from the measured frequency response spectra, two approaches exist toidentify the parameters m, c and k of the Voigt models. The models can be identied by using the primarystiffness k determined from the static stiffness experiment, then m k=o2n and c 2mzon. Or, the isolatedmass m measured on a weight scale can be used, then k mo2n and c 2mzon. A Voigt model identied bythe rst approach will be referred to as Voigt Model Kthe stiffness consistent Voigt model, and the Voigtmodel identied by the second approach will be referred to as Voigt Model Mthe mass consistent Voigt

ARTICLE IN PRESS

0 2 4 6 8 10

0

20

40

60

80

100

120

Load (

N)

0 1 2 3

0

20

40

60

80

Load (

N)

0 1

0

50

100

150

200

Lo

ad (

N)

Deflection (mm)

Deflection (mm)

1.5 2.50.5

1.5

Deflection (mm)

0.5

Unloading

Loading

Unloading

Loading

Unloading

Loading

a

b

c

Fig. 8. Axial loads vs. deections of (a) bubble mount; (b) plate mount; (c) rubber stud.

Table 6

Static stiffness of rubble mounts

Mount type Bubble mount Plate mount Rubber stud

Linear static stiffness (kN/m) 14.013 23.151 54.507

J. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322824

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2825model. The corresponding parameters of the identied Voigt models for each rubber mount are listed inTable 8. The frequency response of Voigt model K, Voigt Model M and experiment are shown in Fig. 11. Thefrequency response of each Voigt Model K are larger in magnitude than the measured frequency response,while the frequency response of each Voigt Model M are smaller in magnitude than the measured frequencyresponse. The frequency response of each Voigt Model M more closely match the experimentally measuredfrequency response except in the low frequency range. This is due to the fact that the Voigt Model M canmatch the natural frequency and damping ratio but not the static stiffness. Alternatively, the Voigt Model Kcan match the static stiffness, natural frequency and damping ratio. However, the frequency response is shiftedhigher in magnitude than the measured frequency response across the entire frequency range.

6.2. Parameter identification as M2V models

In this section the method developed by Zhang and Richards [17] is utilized to identify MV models for thethree rubber isolators. The identied parameters of the MV models are listed in Table 9 (Trial 1) where all theMaxwell elements are identied as Type Cstiffness dominant. Consequently, since the models values for k

Fig. 9. Dynamic experiment system.are chosen to match the experimentally measured static stiffness values, the models identied values for k1result in sums k k1 equal to values such that the models natural frequencies match the estimated naturalfrequencies from experiment. Also, the identied damping coefcients c equal values such that the modelsdamping ratios match the estimated damping ratios from experiment. Therefore, the MV models match notonly natural frequency and damping ratio but also static stiffness. The measured and identied frequencyresponse are shown in Fig. 12. The frequency response of MV Models match much closer to theexperimentally measured frequency response compared to those of the respective Voigt Model M in the lowfrequency range. However, the accuracy of the MV Models frequency response are similar to those of therespective Voigt Model M near and above the peak frequency.From Ref. [17], it is shown that the MVModels frequency response will increase in magnitude as the value

of the primary damper c decreases. Therefore, in order to more closely match the models frequency responsewith experiment in the range near the natural frequency, another group of parameters are identied as listed inTable 10 (Trial 2). Here the identied Maxwell elements belong to Type B and the primary dampingcoefcients c are negative. Since the Type B elements contribute damping to the system, the combined dynamiceffect of the Type B element and primary damper are similar to the primary damper of MV models in Table 9(Trial 1). The Type B elements also contribute stiffness to the system with a dynamic effect similar to the TypeC elements of MV models in Table 9 (Trial 1). The frequency response of these new MV Models areillustrated in Fig. 13. As shown, the curves match very well in the range near the natural frequency, but not aswell in the low frequency range.

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mag

nitude (

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frequency (Hz)

frequency (Hz)

frequency (Hz)

a

b

c

Fig. 10. Frequency response spectra of (a) bubble mount; (b) plate mount; (c) rubber stud.

Table 7

Natural frequency and damping ratio of rubble mounts

Part number Bubble mount Plate mount Rubber stud

Natural frequency (Hz) 29.79 36.54 89.01

Damping ratio 0.0893 0.0896 0.1515

J. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322826

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FRFof Voigt

Model K

FRF of Voigt

Model M

measured FRF

a

b

c

Fig. 11. Frequency response spectra of (a) bubble mount; (b) plate mount; (c) rubber stud.

Table 8

Parameters of identied Voigt models

Part number Bubble mount Plate mount Rubber stud

Voigt model K m (kg) 0.4001 0.4392 0.1743

k (kN/m) 14.013 23.151 54.507

c (N s/m) 13.3728 18.07 29.5299

Voigt model M m (kg) 1.041 1.041 1.041

k (kN/m) 36.461 54.872 325.630

c (N s/m) 34.7954 42.8289 176.4115

J. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2827

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b

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Fig. 12. Frequency response spectra of (a) bubble mount; (b) plate mount; (c) rubber stud (Trial 1).

Table 9

Parameters of identied MaxwellVoigt models (Trial 1)

Isolator type Bubble mount Plate mount Rubber stud

m (kg) 1.041 1.041 1.041

k (kN/m) 14.013 23.151 54.507

c (N s/m) 25.809 31.8765 153.3613

k1 (kN/m) 22.2738 31.4694 267.825

c1 (N s/m) 1588.57 1729.11 10070.89

t1 (ms) [Type] 71.32 [C] 55.09 [C] 37.60 [C]

J. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322828

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Table 10

Parameters of identied MaxwellVoigt models (Trial 2)

Isolator type Bubble mount Plate mount Rubber stud

m (kg) 1.041 1.041 1.041

k (kN/m) 14.013 23.151 54.507

c (N s/m) 40 27 125k1 (kN/m) 28.677 36.951 324.86

c1 (N s/m) 245.7311 318.4438 934.1629

t1 (ms) [Type] 8.57 [B] 8.62 [B] 2.88 [B]

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measured FRF

FRF of M-Vmodel

magnitude (

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magnitude (

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Fig. 13. Frequency response spectra of (a) bubble mount; (b) plate mount; (c) rubber stud (Trial 2).

J. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2829

6.3. Parameter identification of M M V models by constraint optimization

The results of Section 6.2 illustrate that the MV models are incapable of matching the measuredfrequency responses well over the entire frequency range. From Section 3, recall the case where the MVmodel does not match the frequency response of the system consisting of Type B and Type C Maxwellelements. Therefore, the method developed in Section 4 is used to identify the parameters of rubber isolatormodels with two Maxwell elements (MMV models). The resulting parameters are listed in Table 11.If the resulting Maxwell elements of these models belong to Type B and Type C, this will be consistentwith the conclusion in Section 4, i.e., the two Maxwell elements in the identied MMV model thataccurately simulate the static and dynamic behavior of Maxwell systems should belong to Type B andType C.Since the values of t1 and t2 in Table 11 are very different for all three models, it is easy to conclude that the

two Maxwell elements in each model belong to different Types. Also, note the negative values identied for the

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322830primary damper c. Consequently, since the models overall damping must be positive, there must be at leastone Maxwell element in each of the MMV models that belong to Type A or B since only Type A and Belements contribute damping to the system (Type C elements are stiffness dominant and therefore do not adddamping). Since the MV models in Table 9 present correct natural frequencies and damping ratios and theidentied Maxwell elements belong to Type C, stiffness dominant, the primary damper c provides all thedamping of the mounting system. Therefore, the combined damping effect of the Type A or B element and theprimary damper c of the MMV models in Table 11 should be equivalent to the primary damper c ofMV Models in Table 9. If the Maxwell element with t1 is damping dominant (Type A), the sum of c and c1 inTable 11 should be similar to the value of c in Table 9. By comparing the data in Table 9 and 11, it is foundthat the sum of c and c1 in Table 11 are much larger than to the value of c in Table 9 for all three models.Therefore, it is concluded that the Maxwell element with t1 in each model is not Type A but Type B. Since t2 ismuch larger than t1, the second Maxwell element should belong to Type C for all three models. This resultmatches the conclusion in Section 4 where the two Maxwell elements identied in the MMV model belongto Type B and Type C.Since Type B elements also contribute stiffness, the combined stiffness effect of the Type B and C

elements of the MMV models in Table 11 should be equivalent to the Type C elements of MV models inTable 9. Since the stiffness values k2 of the stiffness dominant elements in Table 11 are smaller than thestiffness values k1 in Table 9, the Maxwell element with t1 should belong to Type B or C in order to contributestiffness to the systems. Also, the sum of k1 and k2 in Table 11 are obviously larger than the stiffness values k1in Table 9. Therefore, at least one of the Maxwell elements of MMV models is not Type C. This againdemonstrates that the two identied Maxwell elements of the MMV models in Table 11 belong to Type Band Type C.The frequency response of identied MMV models and experimental system are shown in Fig. 14. As

shown, the frequency response of models match much closer to the experimentally measured frequencyresponse compared with the results from Section 6.2.

Table 11

Parameters of identied Maxwell models

Part number Bubble mount Plate mount Rubber stud

m (kg) 1.041 1.041 1.041

k (kN/m) 14.013 23.151 54.507

c (N s/m) 55.4890 29.1481 70.1778k1 (kN/m) 64.4418 45.2064 389.694

c1 (N s/m) 87.8077 70.9268 239.462

t1 (ms) [Type] 1.36 [B] 1.57 [B] 0.61 [B]k2 (kN/m) 18.2317 26.0506 224.109

c2 (N s/m) 2297.39 2402.82 15507.4

t2 (ms) [Type] 126.01 [C] 92.24 [C] 69.20 [C]

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 28142832 2831-2

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a7. Conclusions

In this study, a parameter identication method based on constraint optimization is used for identifying theparameters of Maxwell models that have two or more Maxwell elements in a single mass isolation system bytting the models to measured frequency response spectra. The identication method was validated by severalanalytical examples. These studies reveal that a Maxwell model having only one Maxwell element can simulatethe dynamic characteristics of a Maxwell system having two Maxwell elements as long as one is not Type Band the other is not Type C. These analytical studies also conclude that a Maxwell model having two Maxwellelements (one Type B and one Type C) can simulate the dynamic characteristics of a Maxwell system havingmore than two Maxwell elements.

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10

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magnitude (

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/N)

0 50 100 150

10-2

magnitude (

mm

/N)

frequency (Hz)

frequency (Hz)

frequency (Hz)

b

c

Fig. 14. Frequency response spectra of (a) bubble mount; (b) plate mount; (c) rubber stud.

Experiments on real elastomeric isolators are conducted with three different rubber isolators subjected toboth static and dynamic experiments. For all three rubber isolators, it is shown that although identied

Proceedings of Noise-Con 98, 1998, pp. 391396.

ARTICLE IN PRESSJ. Zhang, C.M. Richards / Mechanical Systems and Signal Processing 21 (2007) 281428322832[15] W.K. Hong, H.C. Kim, Performance of a multi-story structure with a resilient-friction base isolation system, Computer and

Structures 82 (2004) 22712283.

[16] M. Sjoberg, L. Kari, Non-linear behavior of a rubber isolator system using fractional derivatives, Vehicle System Dynamics 37 (2002)

217236.

[17] J. Zhang, C.M. Richards, Dynamic analysis and parameter identication of a single mass elastomeric isolation system using

MaxwellVoigt model, ASME Journal of Vibration and Acoustics 128 (2006) 713721.

[18] R.K. Brayton, S.W. Director, G.D. Hachtel, L. Vidigal, A new algorithm for statistical circuit design based on quasi-Newton

methods and function splitting, IEEE Trans. Circuits and Systems CAS-26 (1979) 784794.

[19] /http://www.novibes.com/S.[20] C.M. Harris, Shock and Vibration Handbook, third ed., McGraw-Hill, New York, 1988.stiffness consistent Voigt models can match the static stiffness, natural frequency and damping ratio, theycannot match the measured frequency response. Meanwhile, identied mass consistent Voigt models matchmeasured natural frequencies and damping ratios, although they cannot match the measured static stiffnessand frequency response. Identied Maxwell models having only one Maxwell element can match the measurednatural frequency, damping ratio and static stiffness, but cannot match the measured frequency responsecurves well. However, using the method developed in this article, identied Maxwell models having twoMaxwell elements can accurately represent the measured static and dynamic characteristics of real elastomericisolation systems.From the static stiffness experiments in this study, two of the three rubbers isolator exhibit nonlinear static

stiffness. Since all dynamic experiments in this study are conducted under small vibration amplitudeconditions, further work in this area will focus on large vibration amplitude.

References

[1] D.H. Cooper, Method of applying the results of dynamic testing to rubber anti-vibration systems, Transactions of the Institution of

the Rubber Industry 35 (1959) 1661770.

[2] R. G. Dejong, G. E. Ermer, C. S. Paydenkar, and T. M. Remtema, High frequency dynamic properties of rubber isolation elements,

Proceedings of Noise-Con 98, 1998, pp. 383390.

[3] T.R. Lin, N.H. Farag, J. Pan, Evaluation of frequency dependent rubber mount stiffness and damping by impact test, Applied

Acoustics 66 (2005) 829844.

[4] D.J. Thompson, W.J. Van Vliet, J.W. Verheij, Developments of the indirect method for measuring the high frequency dynamic

stiffness of resilient elements, Journal of Sound and Vibration 213 (1) (1998) 169188.

[5] S. Kim, R. Singh, Multi-dimensional characterization of vibration isolators over a wide range of frequencies, Journal of Sound and

Vibration 245 (2001) 877913.

[6] J.H. Lee, K.J. Kim, Treatment of frequency-dependent complex stiffness for commercial multi-body dynamic analysis programs,

Mechanics of Structures and Machines 30 (4) (2002) 527541.

[7] W. Flugge, Viscoelasticity, second revised edition, Springer, New York, 1975.

[8] C.L. Kirk, Non-linear random vibration isolators, Journal of Sound and Vibration 124 (1988) 157182.

[9] N. Chandra Shekhar, H. Hatwal, A.K. Mallik, Response of non-linear dissipative shock isolators, Journal of Sound and Vibration

214 (1998) 589603.

[10] N. Chandra Shekhar, H. Hatwal, A.K. Mallik, Performance of non-linear isolators and absorbers to shock excitation, Journal of

Sound and Vibration 227 (1999) 293307.

[11] C.M. Richards, R. Singh, Characterization of rubber isolator nonlinearities in the context of single- and multi-degree-of-freedom

experimental systems, Journal of Sound and Vibration 247 (5) (2001) 807834.

[12] Y.Q. Ni, J.M. Ko, C.W. Wong, Identication of nonlinear hysteretic isolators from periodic vibration tests, Journal of sound and

Vibration 217 (4) (1998) 737756.

[13] R.Y. Tan, M.C. Huang, System identication of a bridge with lead-rubber bearings, Computer and Structures 74 (2000) 267280.

[14] C. M. Richards, and R. Singh, Identication of nonlinear properties of rubber isolators using experimental and analytical methods,

Parameter identification of analytical and experimental rubber isolators represented by Maxwell modelsIntroductionProblem formulationIdentification of Maxwell systems using M-V modelsParameter identification using constraint optimizationExperimental setup and resultsStatic stiffness experimentFrequency response experiment

Parameter identificationVoigt model parameter identificationParameter identification as M V modelsParameter identification of M-M-V models by constraint optimization

ConclusionsReferences